A360906 Numbers with the same number of cubefree divisors and 3-full divisors.

1, 16, 81, 384, 625, 640, 896, 1296, 1408, 1664, 2176, 2401, 2432, 2944, 3712, 3968, 4374, 4736, 5248, 5504, 6016, 6784, 7552, 7808, 8576, 9088, 9216, 9344, 10000, 10112, 10624, 10935, 11392, 12416, 12928, 13184, 13696, 13952, 14464, 14641, 15309, 16256, 16768

Offset

1,2

Comments

Numbers k such that A073184(k) = A190867(k).

Numbers whose largest cubefree divisor (A007948) and cubefull part (A360540) have the same number of divisors (A000005).

If k and m are coprime terms, then k*m is also a term.

The characteristic function of this sequence depends only on prime signature.

1 is the only cubefree (A004709) term.

Includes the 4th powers of squarefree numbers (1 and A113849).

The 4th powers of primes (A030514) are the only terms that are prime powers (A246655).

Numbers of the for m*p^(3*2^k+1), where m is squarefree, p is prime, gcd(m, p) = 1 and omega(m) = k, are all terms. In particular, this sequence includes numbers of the form p^7*q, where p != q are primes (A179664), and numbers of the form p^13*q*r where p, q, and r are distinct primes.

The corresponding numbers of cubefree (or 3-full) divisors are 1, 3, 3, 6, 3, 6, 6, 9, 6, 6, 6, 3, 6, 6, ... .

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Mathematica

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ (Min[#, 3] & /@ (e + 1)) == Times @@ (Max[#, 1] & /@ (e - 1))]; q[1] = True; Select[Range[10^4], q]

Prog

(PARI) is(k) = {my(e = factor(k)[, 2]); prod(i = 1, #e, min(e[i] + 1, 3)) == prod(i = 1, #e, max(e[i] - 1, 1)); }

Crossrefs

Cf. A000005, A004709, A007948, A073184, A190867, A246655, A360540, A360902.

Subsequences: A030514, A113849, A179664, A360907.

Sequence in context: A351602 A134606 A343324 * A343284 A108941 A153157

Adjacent sequences: A360903 A360904 A360905 * A360907 A360908 A360909

Keyword

nonn

Author

Amiram Eldar, Feb 25 2023

Status

approved